Abdullah Ali Sivas
Applied Mathematics, University of Waterloo
Efficient Preconditioning of Hybridizable/Embedded Discontinuous Galerkin for the Navier-Stokes Equations
In this talk, we will discuss preconditioning techniques for Hybridizable/Embedded Discontinuous Galerkin (HDG/EDG) methods for the approximate solution of partial differential equations (PDEs). For this we will first explain the advantages of HDG/EDG methods over other finite element methods for PDEs, such as the Continuous Galerkin (CG) and the Discontinuous Galerkin (DG) methods. For example, the HDG/EDG methods have less globally coupled degrees of freedom than a DG method on the same grid, and are more stable than CG methods for advection-dominated flows.
For engineering applications, HDG/EDG discretizations typically result in large linear systems of equations which are challenging to solve. To solve these linear systems, the most robust technique is to use direct solution methods. These methods are resilient against numerical errors and can find the solution up to machine precision as long as the coefficient matrix is non-singular. Unfortunately, these methods are expensive and impractical to use for very large linear systems. Therefore, we will focus on iterative solution methods. These are not as robust as direct methods, but they are significantly cheaper. We can solve the robustness problem using preconditioners. The purpose of preconditioners is to change the coefficient matrix to one that is easier to solve using iterative methods. Usually the performance of a preconditioner depends on the mesh size: as one refines the mesh to get a more accurate solution, the performance of the preconditioner degrades. Optimal preconditioners are a special kind of preconditioner which do not suffer from this phenomenon. They have to be found on a case-by-case basis and only a handful of them are known.
We will demonstrate known optimal preconditioners for HDG/EDG discretizations of the Poisson and Stokes problems. We also will compare cases of using no preconditioner, a non-optimal preconditioner and an optimal preconditioner for these. There are no known optimal preconditioners for discretizations of the Navier-Stokes equations, although non-optimal preconditioners which perform well for low Reynolds numbers exist for some discretizations. We will review some of the challenges related to preconditioning discretizations of the Navier-Stokes equations. Ultimately, our goal is to find an efficient, scalable and ideally an optimal preconditioner for the Navier-Stokes equations.